0 Introduction to Game Theory Lirong Xia Voting: manipulation - PowerPoint PPT Presentation

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Introduction to Game Theory Lirong Xia

Voting: manipulation (ties are broken alphabetically) > > YOU > > Plurality rule Bob > > Carol > >

What if everyone is incentivized to lie? > > YOU Plurality rule > > Bob > > Carol

History of Game Theory Ø On the Theory of Games of Strategy. Mathematische Annalen, 1928. • John von Neumann 4

Nobel Prize Winners Ø 1994: • Nash (Nash equilibrium) • Selten (Subgame pefect equilibrium) • Harsanyi (Bayesian games) Ø 2005 • Schelling (evolutionary game theory) • Aumann (correlated equilibrium) Ø 2014 • Jean Tirole 5

A game of two prisoners Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect Ø Players: Ø Strategies: { Cooperate, Defect } Ø Outcomes: {( -2 , -2 ), ( -3 , 0 ) , ( 0 , -3 ), ( -1 , -1 )} Ø Preferences: self-interested 0 > -1 > -2 > -3 : ( 0 , -3 ) > ( -1 , -1 ) > ( -2 , -2 ) > ( -3 , 0 ) • : ( -3 , 0 ) > ( -1 , -1 ) > ( -2 , -2 ) > ( 0 , -3 ) • 6 Ø Mechanism: the table

Formal Definition of a Game Strategy Profile D Mechanism * R 1 s 1 s 2 * R 2 Outcome … … * R n s n • Players: N ={ 1 ,…,n } • Strategies (actions): - S j for agent j, s j ∈ S j - ( s 1 ,…, s n ) is called a strategy profile. • Outcomes: O • Mechanism f : Π j S j → O • Preferences: total preorders (full rankings with ties) over O • often represented by a utility function u i : O → R 7

A game of plurality elections > > Plurality rule YOU > > Bob > > Carol • Players: { YOU, Bob, Carol } • Outcomes: O = { , , } • Strategies: S j = Rankings( O ) • Preferences: See above • Mechanism: the plurality rule 8

Solving the game Ø Suppose • every player wants to make the outcome as preferable (to her) as possible by controlling her own strategy (but not the other players’) Ø What is the outcome? • No one knows for sure • A “stable” situation seems reasonable Ø A Nash Equilibrium (NE) is a strategy profile ( s 1 ,…, s n ) such that • For every player j and every s j ' ∈ S j , f ( s j , s - j ) ≥ j f ( s j ' , s - j ) or equivalently u j ( s j , s - j ) ≥ u j ( s j ' , s - j ) • s - j = ( s 1 ,…, s j- 1 , s j +1 ,…, s n ) • no single player can be better off by unilateral deviation 9

Prisoner’s dilemma Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect 10

The Game of Chicken Ø Two drivers arrives at a cross road • each can either (D)air or (C)hicken out • If both choose D, then crash. • If one chooses C and the other chooses D, the latter “wins”. • If both choose C, both are survived Column player Dare Chicken ( 0 , 0 ) ( 7 , 2 ) Dare Row player ( 2 , 7 ) ( 6 , 6 ) Chicken 11 NE

A beautiful mind Ø “If everyone competes for the blond, we block each other and no one gets her. So then we all go for her friends. But they give us the cold shoulder, because no one likes to be second choice. Again, no winner. But what if none of us go for the blond. We don’t get in each other’s way, we don’t insult the other girls. That’s the only way we win. That’s the only way we all get [a girl.]” 12

A beautiful mind: the bar game Hansen Column player Blond Another girl ( 0 , 0 ) ( 5 , 1 ) Blond Nash Row player ( 1 , 5 ) ( 2 , 2 ) Another girl Ø Players: { Nash, Hansen } Ø Strategies: { Blond, another girl } Ø Outcomes: {( 0 , 0 ), ( 5 , 1 ) , ( 1 , 5 ), ( 2 , 2 )} Ø Preferences: self-interested 13 Ø Mechanism: the table

Does an NE always exists? Ø Not always (matching pennis game) Column player H T ( -1 , 1 ) ( 1 , -1 ) H Row player ( 1 , -1 ) ( -1 , 1 ) T Ø But an NE exists when every player has a dominant strategy • s j is a dominant strategy for player j, if for every s j ' ∈ S j , for every s - j , f ( s j , s - j ) ≥ j f ( s j ' , s - j ) 1. 14 2. the preference is strict for some s - j

Dominant-strategy NE Ø For player j , strategy s j dominates strategy s j ’, if for every s - j , u j ( s j , s - j ) ≥ u j ( s j ' , s - j ) 1. 2. the preference is strict for some s - j 3. strict dominance: inequality is strict for every s - j Ø Recall that an NE exists when every player has a dominant strategy s j , if • s j dominates other strategies of the same agent Ø A dominant-strategy NE (DSNE) is an NE where • every player takes a dominant strategy • may not exists • if strict DSNE exists, then it is the unique NE 15

Prisoner’s dilemma Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect Defect is the dominant strategy for both players 16

Rock Paper Scissors Ø Actions: {R, P, S} Ø Two-player zero sum game No pure NE Column player R P S ( 0 , 0 ) ( -1 , 1 ) ( 1 , - 1 ) R Row player ( 1 , - 1 ) ( 0 , 0 ) ( - 1 , 1 ) P ( - 1 , 1 ) ( 1 , - 1 ) ( 0 , 0 ) S 17

Rock Paper Scissors: Lirong vs. young Daughter Ø Actions • Lirong: {R, P, S} • Daughter: {mini R, mini P} Ø Two-player zero sum game Daughter mini R mini P No pure NE ( 0 , 0 ) ( -1 , 1 ) R Lirong ( 1 , - 1 ) ( 0 , 0 ) P ( 1 , - 1 ) ( 1 , - 1 ) S 18

Computing NE: Iterated Elimination Ø Eliminate dominated strategies sequentially Column player L M R Row ( 1 , 0 ) ( 1 , 2 ) ( 0 , 1 ) player U ( 0 , 3 ) ( 0 , 1 ) ( 2 , 0 ) D 19

Normal form games Ø Given pure strategies: S j for agent j Normal form games Ø Players: N ={ 1 ,…,n } Ø Strategies: lotteries (distributions) over S j • L j ∈ Lot( S j ) is called a mixed strategy • ( L 1 ,…, L n ) is a mixed-strategy profile Ø Outcomes: Π j Lot( S j ) Column player Ø Mechanism: f ( L 1 ,…, L n ) = p L R • p ( s 1 ,…, s n ) = Π j L j ( s j ) Row ( 0 , 1 ) ( 1 , 0 ) U Ø Preferences: player ( 1 , 0 ) ( 0 , 1 ) D • Soon 20

Preferences over lotteries Ø Option 1 vs. Option 2 • Option 1: $0@50%+$30@50% • Option 2: $5 for sure Ø Option 3 vs. Option 4 • Option 3: $0@50%+$30M@50% • Option 4: $5M for sure 21

Lotteries Ø There are m objects. Obj={ o 1 ,…, o m } Ø Lot(Obj): all lotteries (distributions) over Obj Ø In general, an agent’s preferences can be modeled by a preorder (ranking with ties) over Lot(Obj) • But there are infinitely many outcomes 22

Utility theory • Utility function: u : Obj → ℝ Ø For any p ∈ Lot(Obj) • u ( p ) = Σ o ∈ Obj p ( o ) u ( o ) Ø u represents a total preorder over Lot(Obj) • p 1 > p 2 if and only if u ( p 1 )> u ( p 2 ) 23

Example utility Money Money 0 5 30 5M 30M Utility 1 3 10 100 150 Ø u (Option 1) = u ( 0 ) � 50% + u ( 30 ) � 50%=5.5 Ø u (Option 2) = u ( 5 ) � 100%=3 Ø u (Option 3) = u ( 0 ) � 50% + u ( 30M ) � 50%=75.5 Ø u (Option 4) = u ( 5M ) � 100%=100 24

Normal form games Ø Pure strategies: S j for agent j Ø Players: N ={ 1 ,…,n } Ø (Mixed) Strategies: lotteries (distributions) over S j • L j ∈ Lot( S j ) is called a mixed strategy • ( L 1 ,…, L n ) is a mixed-strategy profile Ø Outcomes: Π j Lot( S j ) Ø Mechanism: f ( L 1 ,…, L n ) = p, such that • p ( s 1 ,…, s n ) = Π j L j ( s j ) Ø Preferences: represented by utility functions u 1 ,…, u n 25

Mixed-strategy NE Ø Mixed-strategy Nash Equilibrium is a mixed strategy profile ( L 1 ,…, L n ) s.t. for every j and every L j ' ∈ Lot( S j ) u j ( L j , L - j ) ≥ u j ( L j ' , L - j ) Ø Any normal form game has at least one mixed- strategy NE [Nash 1950] Ø Any L j with L j ( s j )=1 for some s j ∈ S j is called a pure strategy Ø Pure Nash Equilibrium • a special mixed-strategy NE ( L 1 ,…, L n ) where all strategies are pure strategy 26

Example: mixed-strategy NE Column player H T ( -1 , 1 ) ( 1 , -1 ) H Row player ( 1 , -1 ) ( -1 , 1 ) T Ø ( H @0.5+ T @0.5, H @0.5+ T @0.5) } } Row player’s strategy Column player’s strategy 27

Best responses Ø For any agent j , given any other agents’ strategies L - j , the set of best responses is • BR( L - j ) = argmax sj u j ( s j , L - j ) • It is a set of pure strategies Ø A strategy profile L is an NE if and only if • for all agent j , L j only takes positive probabilities on BR( L - j ) 28

Proof of Nash’s Theorem Ø Idea: Brouwer’s fixed point theorem • for any continuous function f mapping a compact convex set to itself, there is a point x such that f ( x ) = x Ø The setting for n players • The compact convex set: Π j =1 n Lot ( S j ) ! "# $% "# (!) • f : L ji à ($∑ # % "# (!) • * +, - = max(2 + - 3+ , 5 +, − 2 + (-), 0) = improvement if switching to a ji Ø Fixed point L * must be an NE • if not, there exists j s.t. ∑ , * +, (-) >0 • L ji > 0 ⇔ * +, (-) > 0 • Improvement on all support, impossible 29

Computing NEs by guessing supports Ø Step 1. “Guess” the support sets Supp j for all players Ø Step 2. Check if there are ways to assign non-negative probabilities to Supp j s.t. • for all s j , t j ∈ Supp j , u j ( s j , L - j ) = u j ( t j , L - j ) • for all s j , ∈ Supp j , t j ∉ Supp j , u j ( s j , L - j ) ≥ u j ( t j , L - j ) 30